What is the maximum number of electrons in an atom that can have the following quantum numbers? Specify the orbitals in which the electrons would be found. (a) \(n=2, m_{\mathrm{s}}=+\frac{1}{2}\) (b) \(n=4, m_{e}=+1\) (c) \(n=3, \ell=2 ;\) (d) \(n=2, \ell=0, m_{\mathrm{s}}=-\frac{1}{2} ;\) (e) \(n=4\) \(\ell=3, m_{\ell}=-2\)

Electron configuration is akin to a roadmap for electrons within an atom; it showcases the specific organization of electrons across different atomic orbitals based on increasing energy levels. Each electron in an atom has a unique set of quantum numbers that dictate its energy level (), subshell (ℓ), and orbital (mℓ), guiding us to its precise address within an atom.

The building-up (or Aufbau) principle helps to determine electron configuration. According to this principle, electrons fill orbitals starting with the lowest available energy states before moving to higher states. A useful mnemonic for this is the phrase 's before p before d before f', reflecting the order in which subshells are typically filled.

The electron configuration for an atom can be written using noble gas notation as a starting point, followed by the configuration of electrons for the remaining subshells. For example, the electron configuration for calcium (Ca) is [Ar] 4s², which tells us that beyond the configuration of argon, there are two electrons in the 4s subshell.

Atomic orbitals are regions within an atom where there is a high probability of finding an electron. These orbitals are defined by three quantum numbers: the principal quantum number (), azimuthal quantum number (), and magnetic quantum number (mℓ). The principal quantum number, n, designates the size and energy level of the orbital, with higher n-values indicative of larger and higher-energy orbitals.

The azimuthal quantum number, ℓ, defines the shape of the orbital, taking on integer values from 0 to n-1 for a given n. When ℓ is 0, the orbital is a spherical 's' orbital; for ℓ=1, it is a dumbbell-shaped 'p' orbital; ℓ=2 corresponds to cloverleaf-shaped 'd' orbitals; and ℓ=3 are the complex-shaped 'f' orbitals.

We use magnetic quantum numbers, m_ℓ, to pinpoint the orientation of the orbital in three-dimensional space. The value of m_ℓ ranges from -ℓ to +ℓ, including zero. Each value of m_ℓ corresponds to a specific orbital; for example, the s subshell (=0) has only one orbital, while the p subshell (=1) has three orientations: px, py, and pz.

The spin quantum number, represented by m_s, is intrinsic to understanding the behavior of electrons within an atom. This quantum number can take on one of two possible values: either +1/2 or -1/2. This binary scenario lays the foundation for the Pauli exclusion principle, stating that no two electrons can have the same set of four quantum numbers. In essence, the spin quantum number tells us the orientation of the electron's spin, which is a fundamental property similar to an angular momentum.

An orbital can hold a maximum of two electrons, each with opposite spins. The spin states are often visualized as electrons spinning in opposite directions, although in reality, the concept of 'spin' does not imply literal spinning. Nonetheless, this property has critical implications, including the formation of magnetic fields and influencing the electron's magnetic moments. Understanding electron spin is integral to quantum mechanics and helps us grasp more complex phenomena such as electron pairing and the overall stability of atoms.